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### How we define the Lie symmetry of a stochastic differential equation?

In "On Lie-point symmetries for Ito stochastic differential equations", they explain some of the difficulties of doing that and some approaches eg. by Kozlov in "Symmetry of systems of ...
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### Distribution of solution to linear SDE

We follow Integrating factor and homogeneous equation for SDEs., but write the details for this specific case \begin{align*} \mathrm{d}X_t = a(t)X_t \mathrm{d}t + d(t)\mathrm{d}B_t. \end{align*} We ...
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### Given a geometric Brownian motion, obtain a recurrence relation for moment $\mathbb{E}X_k^q$ and deduce $\mathbb{E}_k^q=\alpha^k(ah^{1/2},q)$

Taking the $q$th power and expectation of the Euler-Maruyama scheme for this equation gives $$\mathbb{E}X_{k+1}^q=\mathbb{E}\left(1+a\Delta W_k\right)^q\mathbb{E}X_k^q$$ Using the initial condition, ...
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### How to solve system of stochastic differential equations?

As mentioned in the comments the SDE $$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$ is decoupled and so one can solve it N_{1,t}=N_{1,0}\exp \left(\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{1,t}\...
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This holds as long as $X$ is right-continuous at $0$ (assuming you meant $h$ instead of $2h$ in the denominator). I'll just show it for $d=1$ and $Y_0 = 0$, but the general case works exactly the ...